Convergence-Symmetric Metric Spaces

Abstract

We study some basic properties of spaces which satisfy the usual axioms of a metric space but without the symmetry axiom. We realised that such a study is needed in view of the relatively recent appearance of several papers on natural asymmetric metrics to which the available theories do not apply. One prominent example is Thurston's metric on the Teichm{ü}ller spaces of hyperbolic surfaces of finite type, introduced by Thurston in 1985, with several variants and generalisations. Another asymmetric metric is the earthquake metric, also introduced by Thurston and about which several basic questions remain open. Other asymmetric metrics we consider here include the Funk metric, the Apollonian metric and the left Hausdorff metric. We are particularly interested in questions of completeness and completion and in associated notions of boundary at infinity for these asymmetric metrics. In this paper, after discussing several examples, we prove results such as a Banach fixed point theorem, an Arzel{à}--Ascoli theorem and a Hopf--Rinow theorem adapted to this asymmetric setting. We introduce a property we call ``convergence-symmetry'' which turns out to be crucial in the study of metric completeness of some asymmetric metrics. This property is stronger than a property which was formulated by Herbert Busemann around 1970, and which we call the ``Busemann condition''. Every convergence-symmetric metric space has a unique minimal completion which is convergence-symmetric. This does not hold for spaces satisfying Busemann's condition. Several examples we consider satisfy Busemann's condition but are not convergence-symmetric. We have included throughout the paper a certain number of open questions.